Algorithm 919

Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ϕ-Functions Appearing in Exponential Integrators We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 20 articles )

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  1. Luan, Vu Thai: Fourth-order two-stage explicit exponential integrators for time-dependent PDEs (2017)
  2. Bader, Philipp; Blanes, Sergio; Casas, Fernando; Ponsoda, Enrique: Efficient numerical integration of $N$th-order non-autonomous linear differential equations (2016)
  3. Botchev, Mikhail A.: Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling (2016)
  4. Caliari, Marco; Kandolf, Peter; Ostermann, Alexander; Rainer, Stefan: The Leja method revisited: backward error analysis for the matrix exponential (2016)
  5. Farquhar, Megan E.; Moroney, Timothy J.; Yang, Qianqian; Turner, Ian W.: GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion (2016)
  6. Gaudreault, Stéphane; Pudykiewicz, Janusz A.: An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere (2016)
  7. Li, Dongping; Cong, Yuhao; Xia, Kaifeng: Flexible exponential integration methods for large systems of differential equations (2016)
  8. Wu, Gang; Zhang, Lu; Xu, Ting-ting: A framework of the harmonic Arnoldi method for evaluating $\varphi$-functions with applications to exponential integrators (2016)
  9. González, Cesáreo; Thalhammer, Mechthild: Higher-order exponential integrators for quasi-linear parabolic problems. I: Stability (2015)
  10. Tambue, Antoine; Kemajou Brown, Elisabeth; Mohammed, Salah: A stochastic delay model for pricing debt and equity: numerical techniques and applications (2015)
  11. Cieśliński, Jan L.: Improving the accuracy of the AVF method (2014)
  12. Garcia, Ferran; Bonaventura, Luca; Net, Marta; Sánchez, Juan: Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells (2014)
  13. Cieśliński, Jan L.: Locally exact modifications of numerical schemes (2013)
  14. Loffeld, J.; Tokman, M.: Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs (2013)
  15. Bisetti, Fabrizio: Integration of large chemical kinetic mechanisms via exponential methods with Krylov approximations to Jacobian matrix functions (2012)
  16. Blanes, S.; Ponsoda, E.: Time-averaging and exponential integrators for non-homogeneous linear IVPs and BVPs (2012)
  17. Geiger, Sebastian; Lord, Gabriel; Tambue, Antoine: Exponential time integrators for stochastic partial differential equations in 3D reservoir simulation (2012)
  18. Tokman, M.; Loffeld, J.; Tranquilli, P.: New adaptive exponential propagation iterative methods of Runge-Kutta type (2012)
  19. Al-Mohy, Awad H.; Higham, Nicholas J.: Computing the action of the matrix exponential, with an application to exponential integrators (2011)
  20. Tokman, M.: A new class of exponential propagation iterative methods of Runge-Kutta type (EPIRK) (2011)